Theorem spw 2061

Description: Weak version of the specialization scheme sp 2225. Lemma 9 of [KalishMontague] p. 87. While it appears that sp 2225 in its general form does not follow from Tarski's FOL axiom schemes, from this theorem we can prove any instance of sp 2225 having mutually distinct setvar variables and no wff metavariables (see ax12wdemo 2176 for an example of the procedure to eliminate the hypothesis). Other approximations of sp 2225 are spfw 2060 (minimal distinct variable requirements), spnfw 2006 (when 𝑥 is not free in ¬ 𝜑), spvw 2008 (when 𝑥 does not appear in 𝜑), sptruw 1833 (when 𝜑 is true), spfalw 2007 (when 𝜑 is false), and spvv 2015 (where 𝜑 is changed into 𝜓). (Contributed by NM, 9-Apr-2017.) (Proof shortened by Wolf Lammen, 27-Feb-2018.)

Hypothesis

Ref	Expression
spw.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
spw	⊢ (∀𝑥𝜑 → 𝜑)

Distinct variable groups:   𝑥,𝑦   𝜓,𝑥   𝜑,𝑦

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem spw

Step	Hyp	Ref	Expression
1		ax-5 1937	. 2 ⊢ (¬ 𝜓 → ∀𝑥 ¬ 𝜓)
2		ax-5 1937	. 2 ⊢ (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑)
3		ax-5 1937	. 2 ⊢ (¬ 𝜑 → ∀𝑦 ¬ 𝜑)
4		spw.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
5	1, 2, 3, 4	spfw 2060	1 ⊢ (∀𝑥𝜑 → 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209  ∀wal 1565

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1807

This theorem is referenced by:  hba1w  2076  19.8aw  2079  exexw  2080  spaev  2081  ax12w  2174  rspw  3248  reldisj  4419  ralidmw  4482  dtruALT2  5342  bj-ssblem1  37165  bj-ax12w  37189  eu6w  43300